Download Student Overview Sheet Standards GC

Unit 5 Circles – Student Overview Sheet
Standards
G-C-1 Prove that all circles are similar.
G-GPE-1 Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an equation.
G-C-5 Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
G-F-TF-1 Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
G-F-TF-2 Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
G-C-2 Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter
are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects
the circle.
G-CO-13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle
G-C-3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle
G-GMD-1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal
limit arguments.
G-CO-1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Learning Targets
1. I can prove that all circles are similar.
2. I can derive the equation of a circle of given center and radius using the Pythagorean
Theorem.
3. I can complete the square to find the center and radius of a circle given by an equation.
4. I can derive the fact that the length of the arc intercepted by an angle is proportional to
the radius by using similarity
5. I can define the radian measure of the angle as the constant of proportionality
6. I can derive the formula for the area of a sector
7. I can define radian measure of an angle in terms of arc length on the unit circle.
8. I can explain how trigonometric functions are related to the coordinate plane using the
unit circle.
9. I can identify and describe relationships among inscribed angles.
10. I can identify and describe relationships among radii.
11. I can identify and describe relationships among chords
12. I can construct an equilateral triangle.
13. I can construct a square.
14. I can construct a regular hexagon inscribed in a circle
15. I can construct the inscribed circle of a triangle
16. I can construct the circumscribed circle of a triangle
17. I can prove properties of angles for a quadrilateral inscribed in a circle
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Give an informal argument for the formulas for the circumference and area of a circle.
Give an informal argument for the formulas for the volume of a cylinder, pyramid and cone.
Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
Define angle, circle, perpendicular line, parallel line, and line segment.
Understand the notions: point, line, distance along a line, and distance around a circular arc.
Demonstrators
A student started to create a pie chart to track the radio sales at a particular store. Unfortunately,
he only completed the sector that represented radios without CD players, as seen in the figure
below. Answer the following questions and defend your reasoning.
Describe how angles can be measured in radians using the following terms: unit circle, arc
length, angle, radian
Write a paragraph that describes the relationship between the following terms: unit circle, radius,
coordinate pair, sine value, cosine value, tangent value, and angle
Describe how you could construct a right triangle in a circle that does not use a diameter as one
of its sides. Find x and y.
In the figure below…
1)
2)
3)
Students can reproduce and explain constructions as listed above in a variety of media
Students use traditional construction techniques to construct each circle and can provide a
commentary on why the process works.
Students use technology (Geometer’s Sketchpad, Geogebra) to explore angle measures in an
inscribed quadrilateral, make hypotheses, and prove these statements in a paragraph or twocolumn proof.